EXTENSIONS OF HOLOMORPHIC MAPS

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 172, October 1972 EXTENSIONS OF HOLOMORPHIC MAPS BY PETER KIERNAN ABSTRACT. Several generalizations of the big Picard theorem are obtained. We consider holomorphie maps / from X A into M C Y. Under various assumptions on X, A, and M we show that / can be extended to a holomorphie or meromorphic map of X into Y. 1. Introduction. The big Picard theorem says that any holomorphie map / from the punctured disk D into the Riemann sphere Pj(C) which omits three points can be extended to a holomorphie map /: D» P,(C). Kobayashi [3] has obtained the following generalization of this. Let A be a closed submanifold of the complex manifold X and let M be a hyperbolically imbeddedi1) subspace (definition below) of the complex space Y. Then any holomorphie map /: X -A > M can be extended to a holomorphie map f: X Y. The purpose of this paper is to consider further generalizations of this result. In particular, it will be shown that if A has singularities, then in general / can only be extended to a meromorphic map. However, if the singularities of A are normal crossings, then / can be extended to a holomorphie map into Y. 2. Definitions. Our main tool for investigating the extension of holomorphie maps will be the Kobayashi pseudo-distance dm which is associated with the complex space M. For completeness, we shall give the definition and a summary of some of the main properties of d which we use. The reader can consult [3] for further details. Let p and q be points in the complex space M. By a chain a from p to q, we mean a sequence of points p = p., p.,, p, = q in M, points a.,, a, in the unit disk D = \z C \z\ < li and holomorphie maps /.,,/, of D into M with f.(0)=p.. and / (a.) = p.. The length ]a of a is defined by k fe 1 + \a.\ \a\ = V d(0, a) = Y log TT, 1 - aa 1 Received by the editors February 1, 1971 and, in revised form, October 20, AMS(MOS) subject classifications (1970). Primary 32H25, 32H20. Key words and phrases. Big Picard theorem, hyperbolic, Kobayashi pseudo-distance, Satake compactification, symmetric bounded domain. (1) Several characterizations of hyperbolically imbedded spaces and another application of Theorem 2 are given in a recent paper by this author. The paper is entitled Hyperbolically imbedded spaces and the big Picard theorem and has been submitted to Mathematische Annalen. Copyright 1973, American Mathematical Society 347

2 348 PETER KIERNAN [October where d is the Poincaré-Bergman distance on D. It is given by the metric ds = -dzdz. (l- z 2)2 We set dm(p, q) = inf. a, where A is the set of all chains from p to q. It is easy to see that dm is a pseudo-distance on M. If d is a (complete) metric, we say that M is (complete) hyperbolic. Some of the properties of this pseudo-distance which shall be useful to us are (1) dn is the Poincare-Bergman distance on the unit disk D. (2) Let D = \z C 0 < \z\ < l\. Then dp* is a complete metric corresponding to the hermitian metric ds2 =- z 2(log z 2)2 dzdz. Thus D is complete hyperbolic. However, if y = \z = relt\0 <_t <C 2z7J, then the diameter of y converges to 0 as r converges to 0. (3) If /: M * N is holomorphie, then / is distance decreasing. That is, if p, q M then dn(f(p), f(q)) <dm(p, q). (4) M is (complete) hyperbolic iff the universal covering space of M is (complete) hyperbolic. Let V be a complex space and let M be a relatively compact, hyperbolic complex subspace of Y. We say that M is hyperbolically imbedded in Y ii either of the following (equivalent) conditions are satisfied: (i) If p and q are boundary points of M and if \p i and \q! are sequences in M such that p > p, q > q, and d (p, q )» 0, then p - q. (ii) If p is a boundary point of M and U is a neighborhood of p in Y, there exists a neighborhood V of p in V such that V C U and the distance between M n (V - U) and M n V with respect to zz\. is positive. Intuitively, a relatively compact subspace M is hyperbolically imbedded in y if M is hyperbolic and if the "distance" between any two distinct boundary points is greater than zero. The following are some examples of hyperbolically imbedded spaces. (5) Let M be a compact hyperbolic manifold. Then M is hyperbolically imbedded in y = M. (6) Let y = P.(C) and M = P,(C) - {three points}. M is covered by the unit disk and therefore it is complete hyperbolic. Since M has only isolated boundary points, M is hyperbolically imbedded in Y. (7) If M. and M- are hyperbolically imbedded in Y. and Y. respectively, then M.xM. is hyperbolically imbedded in Y. x Y'. To see this let n.: M, x M - > M. be the projection onto M. and let p = (p, p"), q = iq', q"), ^n = (*V P«^' and!? = ^n' O5 be 3S in (i) ab0ve' If dmlxm2(p - 1J ~* > then Mj^, 9^) >0 and dm7(p^, q'"]) ~»0 since 77j and 772 are distance de-

3 1972] EXTENSIONS OF HOLOMORPHIC MAPS 349 creasing. Thus p' = q' and p" = q" and therefore M.x M is hyperbolically imbedded in Y.xY.. (8) Let P7(C) be the 2-dimensional complex projective space and let Q be a complete quadrilateral. That is, Q is the union of the six projective lines which pass through a set of four points in general position. Then M = P.(C) Q is hyperbolically imbedded in P,(C); see [3, p. 92}. Finally, we recall the definition of meromorphic map given by Remmert in [6]. A meromorphic mapping / from a complex space X into a complex space y is a correspondence satisfying (i) For each point x X, f (x) is a nonempty compact subset of y. (ii) The graph T, = \(x, y) X x Y\y f(x)\ of / is a connected complex subspace of X x Y with dim T, = dim X. (iii) There exists a dense subset X of X such that f (x) is a single point for each x in X. Remmert has shown that the codimension of the set X X is > 2. Thus if X is a normal space and dim X = 1, then / is actually holomorphie. 3. Extension theorems. The following theorem is the basis for our extension theorems. The proof is essentially the same as Mrs. Kwack's proof of Theorem 3 in [5]j although our statement is slightly more general. The winding number argument used below is due to Grauert and Reckziegel. Theorem 1. Let M be hyperbolically imbedded in Y and let f,: D > M be a sequence of holomorphie maps where D - \z C 0 < \z\ < l\. Let \z,} and \z'a be sequences in D converging to 0 and such that /,(z,')» q Y. Then (i) fk(*k) -* 1- (ii) Any holomorphie map f: D» M extends to a holomorphie map f: D * Y. (iii) fk(0) q. Proof, (i) Assume that /Sz,) ' p /= q and that \z,\ < z. Choose a coordinate neighborhood U of p such that U is an analytic subset of W = \(w1, - -,w ) C" \w. ] h-+ \u' \ 2 < 2\, q $ U, and p = (0,, O). Let pk(t) = z^'1 for 0 < t < In. Then the diameters of the sets pk with respect to dßt are converging to 0. Since M is hyperbolically imbedded in Y and since k is distance decreasing, this implies AfeW' is converging to p. Thus for all sufficiently large k, there exists an annulus R!k centered at 0 such that pk C R'k and f ÍR.) C B = \(w,,w )\ \wa\ + +\wn\ < l\. Let Rk be the largest such annulus. We can assume that either R, is not a punctured disk for any k or that R, is a punctured disk for all k. We consider the former case first. Since /, (z, ) > q 4 U, there exist ak and bk ln Rk with W' <!^jj and I /few^i = lafe(^l = * By taking a subsequence and relabelling, we may assume that f,{a,)~>q'eb and il^bj) * q B. Let o^ and r, be the curves defined by o,(t) = a,elt and r, (t) - b,elt for 0 < ; < 2z7. By the same reasoning used for p,, we see that ís }) ~~* 1 ' and fistk) 7 "

4 350 PETER KIERNAN [October Let fk = (fxk,, / ) in f7.l(u). By rotating the coordinate ball W if necessary, we see that the winding number of the curves fao.) and f At, ) around the point fl(z,) is 0 for all k sufficiently large. By Cauchy's theorem, we have 'kk k A' f,'(z) r dw. r /*u) dz- r f /fe'u) T_ r ^1_ J^ / (z) - f\(zk) ' hl(rk)w^fi(zk) 0. Thus,f VU) «fe.f '*'U) l<fe-2,n(p-n) where N and P are the number of zeros and poles of the function / (z) f}(z,) on the annulus R,. This is a contradiction since /V > 0 and P = 0. If R, is a punctured disk for all k, then the boundary of R, is r, where b, and rfe are as above. Since fár^ C B, /, extends holomorphically to R, with /, (O) in B. Setting o. = 0, the argument used in the preceding paragraph leads to a contradiction. This proves (i). (ii) Let z/ Î be any sequence in D such that /(z^)» q Y. Define /(0) = q. By (i), / is continuous. The Riemann extension theorem now implies that / is holomorphie at 0. (iii) If /,(O) does not converge to q, then without loss of generality we may assume that /, (O) > p q. Since each /, is continuous, there exists a sequence z,i in D such that z, >0 and /,(z,)» p. This contradicts (i). Q.E.D. The following theorem was proven by Kwack in the case when Mis compact and by Kobayashi in the case when A is nonsingular. Theorem 2. Let A be a closed complex subspace of a (nonsingular) complex manifold X. If the singularities of A are normal crossings and if M is hyperbolically imbedded in Y, then any holomorphie map f: X A * M extends to a holomorphie map f: X» y. Proof. Since the singularities of A are normal crossings, we can assume that X = Dx-.-xD = DraxD/= \(wv..., wn, tv, iz) \w.\ < 1 and f;. < lj and that X-A = (D*)*x Dl= \(wv..., wn, tv---, t )\0< \w.\ < 1 and \tf\ < l\. The proof is by induction on n.

5 1972] EXTENSIONS OF HOLOMORPHIC MAPS 351 ** Case 1. X - A D. This is part (ii) of Theorem 1. Case 2. Assume we can extend / when X - A = (D )". We show that this implies we can extend / if X A = (D )" x D. Let w = (w, w ) D" and t = (t,, -, t ) D. We denote the point In 1 i (tz/j,, wn, fj,, f;) D" xdl by (t^, r). Let /: (D*)" x Dl» M be holomorphie and define / : (D )"» M by /,(w) = f(w, t). Since each map / extends to D", we can extend / to a map from D" x D into Y by defining /(zz;, /) = f,(w). By the Riemann extension theorem, it suffices to show that the extended map / is continuous. We assume that / is not continuous at some point, say (w, fj) for simplicity, and then obtain a contradiction. If / is not continuous at (w, 0), then there exists a sequence of points (wk, tk) in (D*)" x (D*)1 with (wk, tk)» (w, 0) and such that the sequence f(w, t )» q 4 f(u>, 0). Define /fe: D * M by f' A\z) = f(wk, ztk/\tk\). Let z^ = \tk\. Since z 0 and fk(z'k ) = /(w*, fe) > c?, part (iii) of Theorem 1 implies that fao) - f(w, 0) > q. But / is continuous for each t and therefore f(w, 0) ' f(w, 0) 4- q. Thus the assumption that / is not continuous is false. Case 3. Assume that / can be extended if X - A = (D*)n x D. We show that this implies that / can be extended if X - A = (D )"*. By the inductive hypothesis, we can extend / to D"+ i(0,, 0)S. The map g: D M defined by g(z) = f(z,, z) extends to D. Define /(0,, O) = g(0). It suffices to show that / is continuous. If / is not continuous-, there exists a sequence (w, t ) = (w,, w, t ) in (D*)n+1 such that f(wk, th) > q 4 f(0,, 0). By applying Theorem 1 with fk(z)= f(zwk /\wk\, tk) and z'k = \wk\, we see that /(0, /*) = /fe(0) -* q. On the other hand, if we apply Theorem 1 with fk(z) = f(ztk/\tk\,...,ztk/\tk\,tk) and z'k=\tk\, we see that /(0, / ) " /(0,, 0) 4 q. This is a contradiction and therefore / extends holomorphically. Q.E.D. The following example shows that if M is not compact, restrictions on the singularities of A are necessary in order to make Theorem 2 remain true. Let M = D x (C - I - 1, IS) C Pj(C) x P,(C). Since both D and C - Í - 1, 1! are hyperbolically imbedded in P.(C), example (7) of 2 implies that M is hyperbolically imbedded in PAC) x Pj(C). Let X = D x D and A =!(z, w)\z = 0 or z = ±w\ and define /: X - A > M by /(z, z ) = (z, w/z). Clearly / does not extend to all of X, since /(0, o) cannot be defined in a continuous way. This example is more or less typical of the general situation. To see this consider the following generalization. Let A be a closed complex subspace of X= D" = Dx xd with (0,, 0) A. Let L denote the tautological line bundle over

6 352 PETER KIERNAN [October P _.(C) (i.e. L is the line bundle associated to the principal C bundle n: C - KO,, 0)S *-P,(C) where n is the natural projection). Then X - A can be 7 n 1 L considered as a subspace of L. If X A is hyperbolically imbedded in L, then the inclusion i: X A > L isa counterexample to the singular version of Theorem 2. Note that in these examples the maps do extend to a meromorphic map. This is always the case. Theorem 3. Let A be a closed complex subspace of a complex space X and let M be hyperbolically imbedded in Y. If f: X A M zs holomorphie, to a meromorphic map f: X > y. then f extends Proof. By a resolution of A in X we shall mean a triple (Z, B, cp) where (i) Z is a nonsingular complex manifold (not necessarily connected), (ii) B is a complex subspace of Z whose singularities are normal crossings, (iii) cp: Z > X is a proper holomorphie map onto X with (p~ (A) = B and such that cp~ is meromorphic. By Hironaka [2], such a resolution always exists (at least locally). Define /: Z - B» M by / = / 4>- By Theorem 2, / extends to a holomorphie map /: Z > y. Thus / extends meromorphically to X by defining / = / 4>~ Q.E.D. We shall now weaken the restrictions on the singularities of A given in Theorem 2. Theorem 4. Let A be a closed complex subspace of a normal complex space X. Assume that (Z, B, </>) zs a resolution of A in X (see proof above) such that for every pair of points p and q in B with cf>(p) = <p(q), there exist sequences \p and \q \ in Z - B with p > p, q» q and dv A<f>(p ), <b(q )) ' 0. // 1 n n n J n X~A'n'*n M is hyperbolically imbedded in Y, then any holomorphie map f: X A > M extends to a holomorphie map f: X > y. Proof. Let /: Z > Y be defined as in the last proof. Since X is normal, it suffices to show that the meromorphic map /= / <p~ is single valued. Since M is hyperbolically imbedded in y and since f\x_a ls distance decreasing, we have J(p) = um f(<pip)) = lim f(<ß(q)) = J(q). This implies that / is single valued. Q.E.D. Remark. If X is nonsingular and if A has only normal crossings, then A obviously satisfies the conditions of Theorem 4. Thus Theorem 4 is a generalization of Theorem 2, but in practice it is not a very satisfactory one because we are usually unable to actually construct a resolution, however, it does not seem very easy to give a condition which depends only on X A and which is not too restrictive.

7 1972] EXTENSIONS OF HOLOMORPHIC MAPS 353 In our final generalization of the Picard theorem, we consider the space M = \(w,,,w ) Cn\w. 4 0 and w. 4 1 for i = 1,, n\. Let [(w, v ),, (w, v )] be "homogeneous" coordinates for Pj(C)x x Pj(C) and let (w, w.,, w ) be homogeneous coordinates for P (C). Define <f>: C"» Pj(C)x -xpj(c) by ( Uj,, w) = [U>r l),, (wn, 1)] and define if,: C" > P (C) by tfj(w,, u>n) «(1, tfj,, w). Thus r/> imbeds M^ in P,(C)x.-.xP.(C) and xb imbeds M in P (C). By (7) of 2, M is hyper- 1 1 ' n n J n - L bolically imbedded in P. (Ox x P.(C) and therefore the results of this section apply in this case. On the other hand, an easy computation shows that the identity map i: M > M does not extend to a holomorphic map i: P,(C)x- -x P,(C) 'P (C). Since the singularities of A = P,(C)x -x P,(C) - M are normal crossings, Theorem 2 implies that M is not hyperbolically imbedded in P (C). However, we still have Theorem 5. Let A be a closed submanifold of a complex manifold X. Then every holomorphic map f: X - A > M extends to a holomorphic map f: X * P (C). Proof. By Theorem 2, / extends to a holomorphic map /: X > P.(C)x xpj(c). We can assume X - Dm = i(zj,...,zj ecm z. < 1 for i' = 1,, m] and that A = \(z ^...,zj Dm\z j = Of. Let /= (/Jt -,/ ) where f'.: X -> P,(C). Without loss of generality, we can assume that /. is given by two holomorphic functions, i.e. f.(z j,, zj = (gf(zj>, z ), h.(z,., z )). Since b{\x_a 4 0, we see that either =0 or h.\a 4 0. This implies that h.(z,, z ) = zkih.(z,, z ) where k. > 0 and i(z,'",z )/0 in X. By tí Tí * t J the same reasoning, we have g^zj,, z^) = Zj'' g (z,, z ) where g.(zj,, z ) 40 in X. This means that / (i.e. if/ f) is given by /(z... z)- il M»(*l',"'*»).f«jn^l»"->*«)\ /vzj,..., zni - ii, zj ^-_-r«,,,.*i j-7-r \ bl{zv...,zn) hn(zv...,zn)j where /. = k. - k.. This implies that / extends to a holomorphic map f: X» P (C). Q.E.D. 4. Applications. Let j) be a bounded symmetric domain in C" and let T be an arithmetically defined, discrete subgroup of the largest connected group of holomorphic automorphisms of 5>. Assume T acts freely on 2). There are two compactifications of ÍD/T. The Satake compactification (5VD _ is the same D.D. as that of Borel-Baily as a topological space. The other compactification UVDp s is that of Pyateckir-Sapiro. In an unpublished work, Borel proved Corollary 1 for (5)/T)B. His proof shows that 5)/T is hyperbolically imbedded in (ÍD/T)B_B. In [4], Kobayashi and Ochiai showed that 2)/T is hyperbolically imbedded in (S/Dp s and proved Corollary 1 for this case when A is nonsingular. Baily has shown that there is a one-one continuous map tfi: (3)/D* * (ÍD/T)*,

8 354 PETER KIERNAN [October but it is not known whether (3)/T)p s is even Hausdorff.(2) Using these remarks and letting M be either (2)/T)R or (3J/r)p <. we have Corollary 1. Let X and A be as in Theorem 4 (or 2). Then any holomorphic map f: X A ' j)/v extends to a holomorphic map f: X > M. If A is a complex subspace of a complex space X, then any holomorphic map f: X A * _D/T extends to a meromorphic map f: X > M. In particular, if f: jj/v ' 3) /T zs holomorphic, then f extends to a meromorphic map f: M > M. theorem. The next corollary can be interpreted as a "generalization" of Liouville's Corollary 2. Let X be a complex manifold with dx = 0 and let A be a closed complex subspace of X. // M is a compact hyperbolic space and f: X A > M is holomorphic, then f is a constant map. (For example, any map from C SO, li into a compact Riemann surface of genus >_2 is a constant map.) If X A is hyperbolic, and V is a group of automorphisms of X A such that X A zs a covering space of (X - A)/T, then (X A)/Y is not compact. Proof. Since M is compact hyperbolic, / extends to a map f: X > M. Since dx = 0 and d is a proper distance, the distance decreasing property of / implies that / is a constant map. Q.E.D. Corollary 3. Let V CP (C) be a complex manifold and let M C P (C) be as J n r ' n n in Theorem 5- Let M - V O M. Let A be a closed complex subspace of a complex space X. Then any holomorphic map f: X A > M extends to a meromorphic map f: X ' V. If X and A are both nonsingular, then f: X» V is holomorphic. Proof. Since M is hyperbolically imbedded in P.(C)x -x P,(C), it is clear that / extends meromorphically. The second statement follows directly from Theorem 5. Q.E.D. The previous corollary generalizes Corollary A of a recent paper of Griffiths [l}. Griffiths has shown that under the proper restrictions on the imbedding of V in P (C), M will have the property that it is covered by a bounded domain in Cm which is topologically a cell. 5. Conclusion. In thi^paper we have considered generalizations of the big Picard theorem. Except for Theorem 5, we have replaced the range space P.(C) Í3 points! by a hyperbolically imbedded complex space, and then we have tried to replace the domain space D by more general spaces. Now we comment on the relevency of the assumption "hyperbolically imbedded". To do this consider the following examples. (i) Let AT =\(z0,zvz2) P2(C)\z040, zx40, z,4zn> z 2 4 ±zqezo/z l\. (2) Borel has recently shown that the two compactifications of 3)/T are equivalent.

9 1972] EXTENSIONS OF HOLOMORPHIC MAPS 355 Define <f>: M ' -* (C - (O, li) x (C -! - 1, l!) by cf>(z0, ZV z2)=(zl/z0, (z2/z0)e-zo/zl). Clearly <f> is a biholomorphism and thus M is complete hyperbolic. However, the map g: D ' M defined by g\z) = (1, z, 2e z) does not extend to a meromorphic map g: D» P2(C). (ii) Let M" =!(z0, Zj, z2) ep2(c) z0 0, Zj 4 0, *, z0> z^ ±z2i. Since the map : D x D M" defined by h(z, w) - (l, z, w/z) does not extend to a holomorphic map h: D x D P.(C), M ' is not hyperbolically imbedded in P2(C>. Define xfi: M" -* (C - JO, 1 j) x (C- {- 1, 10 by ifj(z Q, zp zjm (z./z-, z.z /z ). It is easy to see that tf, is a biholomorphism which extends to a bimeromorphism if/: PAC)» P,(C) x P,(C). Thus Theorem 3 holds for maps into M. The first example shows that some assumptions on the way in which M is contained in Y are necessary if any extension theorem is going to work. The second example indicates that it is not necessary for M to be hyperbolically imbedded in y in order for Theorem 3 to be true. In fact, it is possible that it may be sufficient for M to be a (complete) hyperbolic Zariski open set of a closed algebraic manifold y. It also shows that these last conditions are not strong enough to imply that you can extend holomorphically over a nonsingular subspace. Theorem 5 implies that "hyperbolically imbedded" is not a necessary condition for the nonsingular version of Theorem 2. Finally, the discussion preceding Theorem 5 indicates that strong assumptions such as "hyperbolically imbedded" are probably necessary if Theorem 2 is to hold. REFERENCES 1. P. A. Griffiths, Complex-analytic properties of certain Zariski open sets on algebraic varieties, Ann. of Math. (2) 94 (1971), H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math (2) 79 (1964), MR 33 # S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Appl. Math., no. 2, Dekker, New York, MR 43 # S. Kobayashi and T. Ochiai, Satake compactifications and the great Picard theorem, J. Math. Soc. Japan 23 (1971), M. H. Kwack, Generalization of the big Picard theorem, Ann. of Math. (2) 90 (1969), MR 39 # R. Remmert, Holomorphe und meromorphe Abbildungen komplexer Räume, Math. Ann. 133 (1957), MR 19, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF BRITISH COLUMBIA, VANCOUVER 8, B.C., CANADA